Exotic Preferences for Macroeconomists · 2020. 3. 20. · 320 Backus, Routledge, & Zin (A) zi = 1 z2 = (zo, 1,1) 21=2 Figure 1 A representative event tree. This event tree illustrates

This PDF is a selection from a published volume fromthe National Bureau of Economic Research

Volume Title: NBER Macroeconomics Annual 2004,Volume 19

Volume Author/Editor: Mark Gertler and KennethRogoff, editors

Volume Publisher: MIT Press

Volume ISBN: 0-262-07263-7

Volume URL: http://www.nber.org/books/gert05-1

Publication Date: April 2005

Title: Exotic Preferences for Macroeconomists

Author: David K. Backus, Bryan R. Routledge, StanleyE. Zin

URL: http://www.nber.org/chapters/c6672

Exotic Preferences forMacroeconomists

David K. Backus, Bryan R.Routledge, and Stanley E. ZinNew York University andNBER; Carnegie MellonUniversity; and CarnegieMellon University and NBER

1. Introduction

Applied economists (including ourselves) are generally content tostudy theoretical agents whose preferences are additive over time andacross states of nature. One version goes like this: Time is discrete,with dates t = 0,1,2, At each t > 0, an event zt is drawn from a fi-nite set 2£, following an initial event ZQ. The ^-period history of eventsis denoted by zf = (zo,Zi,... , zf) and the set of possible ^-histories byJ*f. The evolution of events and histories is conveniently illustrated byan event tree, as in Figure 1, with each branch representing an eventand each node a history or state. Environments like this, involvingtime and uncertainty, are the foundation of most of modern macroeco-nomics and finance. A typical agent in such a setting has preferencesover payoffs c(zt) for each possible history. A general set of preferencesmight be represented by a utility function U({c(zt)}). More common,however, is to impose the additive expected utility structure

^>W1)] = Eo EA(Q), (1)

where 0 < /? < 1, p(zf) is the probability of history zf, and u is aperiod/state utility function. These preferences are remarkably parsi-monious: behavior over time and across states depends solely on thediscount factor /?, the probabilities p, and the function u.

Although equation (1) remains the norm throughout economics,there has been extraordinary theoretical progress over the last fiftyyears (and particularly the last twenty-five) in developing alternatives.Some of these alternatives were developed to account for the anoma-lous predictions of expected utility in experimental work. Others arosefrom advances in the pure theory of intertemporal choice. Whatever

320 Backus, Routledge, & Zin

(A)

zi = 1

z2 = (zo, 1,1)

2 1 = 2

Figure 1A representative event tree. This event tree illustrates how uncertainty might evolvethrough time. Time moves from left to right, starting at date t — 0. At each date t, anevent zt occurs. In this example, zt is drawn from the two-element set SE — {1,2}. Eachnode is marked by a box and can be identified from the path of events that leads to it,which we refer to as a history and denote by zt = (zo,Zi,. •. ,zt), starting with an arbi-trary initial node ZQ. Thus the upper right node follows two up branches, z\ — \ and22 = 1, and is denoted z2 = (zo, 1,1). The set HE1 of all possible 2-period histories is there-fore {(z0,1,1), (ZQ, 1,2), (ZQ, 2,1), (ZQ, 2,2)}, illustrated by the far right "column" of nodes.

Exotic Preferences for Macroeconomists 321

their origin, they offer greater flexibility along several dimensions,often with only a modest increase in analytical difficulty.

What follows is a user's guide, intended to serve as an introductionand instruction manual for economists studying problems in whichthe structure of preferences may play an important role. Our goal is todescribe exotic preferences to mainstream economists: preferences overtime, preferences across states or histories, and (especially) combina-tions of the two. We take an overtly practical approach, downplayingor ignoring altogether the many technical issues that arise in specifyingpreferences in dynamic stochastic settings, including their axiomaticfoundations. (References are provided in the appendix for those whoare interested.) We generally assume without comment that prefer-ences can be represented by increasing, (weakly) concave functions,with enough smoothness and boundary conditions to generate inte-rior solutions to optimizations. We focus instead on applications, usingtractable functional forms to revisit some classic problems: consump-tion and saving, portfolio choice, asset pricing, and Pareto optimal allo-cations. In most cases, we use utility functions that are homogeneousof degree 1 (hence invariant to scale) with constant elasticities (thinkpower utility). These functions are the workhorses of macroeconomicsand finance, so little is lost by restricting ourselves in this way.

You might well ask: Why bother? Indeed, we will not be surprisedif most economists continue to use (1) most of the time. Exotic prefer-ences, however, have a number of potential advantages that we believewill lead to much wider application than we've seen to date. One ismore flexible functional forms for approximating features of data—theequity premium, for example. Another is the ability to ask questionsthat have no counterpart in the additive model. How should we makedecisions if we don't know the probability model that generates thedata? Can preferences be dynamically inconsistent? If they are, howdo we make decisions? What is the appropriate welfare criterion? Canwe think of some choices as tempting us away from better ones? Eachof these advantages raises further questions: Are exotic preferencesobservationally equivalent to additive preferences? If not, how do weidentify their parameters? Are they an excuse for free parameters? Dowe even care whether behavior is derived from preferences?

These questions run through a series of nonadditive preferencemodels. In Section 2, we discuss time preference in a deterministic set-ting, comparing Koopmans's time aggregator to the traditional time-additive structure. In Section 3, we describe alternatives to expectedutility in a static setting, using a certainty-equivalent function to


summarize preference toward risk. We argue that the Chew-Dekelclass extends expected utility in useful directions without sacrificinganalytical and empirical convenience. In Section 4, we put time andrisk preference together in a Kreps-Porteus aggregator, which leads toa useful separation between time and risk preference. Dynamic exten-sions of Chew-Dekel preferences follow the well-worn path of Epsteinand Zin. In Section 5, we consider risk-sensitive and robust control,whose application to economics is associated with the work of Hansenand Sargent. Section 6 is devoted to ambiguity, in which agents faceuncertainty over probabilities as well as states. We describe Gilboa andSchmeidler's max-min utility for static settings and Epstein andSchneider's recursive extension to dynamic settings. In Section 7, weturn to hyperbolic discounting and provide an interpretation basedon Gul and Pesendorfer's temptation preferences. The final section isdevoted to a broader discussion of the role and value of exotic prefer-ences in economics.

A word on notation and terminology: We typically denote param-eters by Greek letters and functions and variables by Latin letters. Wedenote derivatives with subscripts; thus, V2 refers to the derivative ofV with respect to its second argument. In a stationary dynamic pro-gramming problem, / is a value function and a prime (') distinguishesa future value from a current value. The abbreviation iid means "inde-pendent and identically distributed," and NID(x, y) means "normallyand independently distributed with mean x and variance y."

2. Time

Time preference is a natural starting point for macroeconomists sinceso much of our subject is concerned with dynamics. Suppose there isno risk and (for this paragraph only) ct is one-dimensional. Preferencesmight then be characterized by a general utility function U({ct}). Acommon measure of time preference in this setting is the marginal rateof substitution between consumption at two consecutive dates (ct andCt+\, say) along a constant consumption path (Q = c for all t). If themarginal rate of substitution is

then time preference is captured by the discount factor


(Picture the slope, — 1//?, of an indifference curve along the 45-degreeline.) If /?(c) is less than 1, the agent is said to be impatient: she requiresmore than one unit of consumption at t + 1 to induce her to give upone unit at t. For the traditional time-additive utility function,

U({ct}) = E A(Q), (2)f=0

/?(c) = p < 1 regardless of the value of c, so impatience is built in andconstant. The rest of this section is concerned with preferences inwhich the discount factor can vary with the level of consumption.

2.1 Koopmans's Time Aggregator

Koopmans (1960) derives a class of stationary recursive preferences byimposing conditions on a general utility function U for a multidimen-sional consumption vector c. Our approach and terminology followJohnsen and Donaldson (1985). Preferences at all dates come from thesame date-zero utility function U. As a result, they are dynamicallyconsistent by construction: preferences over consumption streamsstarting at any future date t are consistent with U. Following Koop-mans, let tc = (cf, ct+i,...) be an infinite consumption sequence startingat t. Then we might write utility from date t — 0 on as

Koopmans's first condition is history-independence: preferences oversequences tc do not depend on consumption at dates prior to t.Without this condition, an agent making sequential decisions wouldneed to keep track of the history of consumption choices to be ableto make future choices consistent with U. The marginal rate of substi-tution between consumption at two arbitrary dates could depend, ingeneral, on consumption at all dates past, present, and future. History-independence rules out dependence on the past. With it, the utilityfunction can be expressed in the form

for some time aggregator V. As a result, choices over \C do not dependon Co- (Note, for example, that marginal rates of substitution betweenelements of \C do not depend on Co) Koopmans's second conditionis future independence: preferences over ct do not depend on t+\C. (In


Koopmans's terminology, the first and second conditions togetherimply that preferences over the present (ct) and future (t+ic) are inde-pendent.) This is trivially true if ct is a scalar, but a restriction on prefer-ences otherwise. The two conditions together imply that utility can bewritten

U(oc) = V[u(co),Ui(ic)]

for some functions V and u, which defines u as a composite commod-ity for consumption at a specific date. Koopmans's third condition isthat preferences are stationary (the same at all dates). The three condi-tions together imply that utility can be written in the stationary recur-sive form

for all dates t. This is a generalization of the traditional utility function(2), where (evidently) V(u, U) = u+ fiU or the equivalent. As in tradi-tional utility theory, preferences are unchanged when we apply a mono-tonic transformation to 17: if L7 = /(If) for / increasing, then we replacethe aggregator V with V{u, 17) = f(V[uJ~l{U)}).

In the Koopmans class of preferences represented by equation (3),time preference is a property of the time aggregator V. Consider ourmeasure of time preference for the composite commodity u. If Ut andut represent U(tc) and u(ct), respectively, then

Ut = V(uh Ut+i) = V[ut, V(ut+U Ut+2)].

The marginal rate of substitution between ut and ut+\ is therefore

woo V2{ut,Ut+l)Vi(Ut+\,Ut+2)MKbM+i = — —— .

Vi(ut,Ut+i)A constant consumption path with period utility u is defined byU = V(u,U), implying U = g(u) = V[u,g(u)] for some function g.(Koopmans calls g the correspondence function.) The discount factor istherefore fi(u) = V2[u,g(u)}. You might verify for yourself that Vi isinvariant to increasing transformations of 17.

In modern applications, we generally work in reverse order: wespecify a period utility function u and a time aggregator V and usethem to characterize the overall utility function 17. Any 17 constructedthis way defines preferences that are dynamically consistent, historyindependent, future independent, and stationary. In contrast to time-additive preferences, discounting depends on the level of utility u.


To get a sense of how this works, consider the behavior of V2. If prefer-ences are increasing in consumption, u must be increasing in c and Vmust be increasing in both arguments. If we consider sequences withconstant consumption, U must be increasing in u, so that

gl(U) = Vi[M,g(M)] + V2[u,g(u)]g1(u) = 1 V_ly^]{u)] > 0-

Since V\ > 0, 0 < V2[",#(")] < 1/ the discount factor is between zeroand one and depends (in general) on u. Many economists imposean additional condition of increasing marginal impatience: V2[u,g(u)] isdecreasing in u, or

V2i[u,g{u)] + V22[u,g(u)}gi{u)

In applications, this condition is typically used to generate stability ofsteady states.

Two variants of Koopmans's structure have been widely used bymacroeconomists. One was proposed by Uzawa (1968), who suggesteda continuous-time version of

(In his model, fi(u) = exp[—S(u)].) Since V21 = 0, increasing marginalimpatience is simply /^(u) < 0 [or 8\{u) > 0]. Another is used byEpstein and Hynes (1983), Lucas and Stokey (1984), and Shi (1994),who generalize Koopmans by omitting the future independence condi-tion. The resulting aggregator is V(c, U), rather than V(u, U), whichallows choice over c to depend on U. If c is a scalar, this is equivalentto (3) [set u(c) = c], but otherwise need not be. An example is

V{c,U) = u(c)+P(c)U,

where there is no particular relationship between the functions u

2.2 Examples

Example 1 (growth and fiscal policy) In the traditional growth model,Koopmans preferences can change both the steady state and the short-run dynamics. Suppose the period utility function is u(c) and the time


aggregator is V(u,W) = u+/?(u)U7, with u increasing and concaveand /?! (w) < 0. Gross output y is produced with capital k using anincreasing concave technology / . The resource constraint is y = f(k) =c + k' + g, where c is consumption, k' is tomorrow's capital stock, and gis government purchases (constant). The Bellman equation is

J(k) = max u[f(k) - k'-g]+P(u[fW ~ *'

The first-order and envelope conditions are

which together imply ]\{k) = f}[u{c)\]\(k')f\{k). In a steady state, 1 =

One clear difference from the traditional model is the role of prefer-ences in determining the steady state. With constant /?, the steady-statecapital stock solves fifi(k) = 1; u is irrelevant. With recursive prefer-ences, the steady state solves J3(u[f(k)- k — g])fi{k) — 1, which de-pends on u through its impact on p. Consider the impact of an increasein g. With traditional preferences, the steady-state capital stock doesn'tchange, so any increase in g is balanced by an equal decrease in c. Withrecursive preferences and increasing marginal impatience, an increasein g reduces current utility and therefore raises the discount factor. Theinitial drop in c is therefore larger than in the traditional case. In theresulting steady state, the increase in g leads to an increase in k and adecline in c that is smaller than the increase in g. The magnitude of thedecline depends on fiv the sensitivity of the discount factor to currentutility. [Adapted from Dolmas and Wynne (1998).]

Example 2 (optimal allocations) Time preference affects the optimal allo-cation of consumption among agents over time. Consider an economywith a constant aggregate endowment y of a single good, to be dividedbetween two agents with Koopmans preferences, represented here bythe aggregators V (the first agent) and W (the second). A Pareto opti-mal allocation is summarized by the Bellman equation:

J(w)=maxV[y-cJ(w')}

c,w'

subject to

W(C,W') > TV.


Note that both consumption c and promised utility w pertain to thesecond agent. If X is the Lagrange multiplier on the constraint, thefirst-order and envelope conditions are

V1[y-c,J(w')]=XW1(c:w')

V2[y - c,J(w')]h(w') + M2(c,v>') = 0

hip) = -A.

If agents' preferences are additive with the same discount factor /?,then the second and third equations imply J\(w')/Ji{w) = W2{c,iv')/Vi[y — c,J(w')] =/?//? = 1: an optimal allocation places the sameweight X — —]\ (w) on the second agent's utility at all dates and prom-ised utility w is constant. If preferences are additive and ft2 > P\ (thesecond agent is more patient), then ]i{w')/]\(w) = fi2/Pi > 1

; a n opti-mal allocation increases the weight over time on the second, morepatient agent and raises her promised utility (w' > w). In the moregeneral Koopmans setting, the dynamics depend on the time aggrega-tors V and W. The allocation converges to a steady state if both aggre-gators exhibit increasing marginal impatience and future utility is anormal good. [Adapted from Lucas and Stokey (1984).]

Example 3 (long-run properties of a small open economy) Small openeconomies with perfect capital mobility raise difficulties with the exis-tence of a steady state that can be resolved by endogenizing the dis-count factor. We represent preferences over sequences of consumptionc and leisure 1 — n with a period utility function M(C, 1 — n) and a timeaggregator V(c, l — n,U) = u(c, 1 - n) + fi(c, 1 - n)U. Let output beproduced with labor using the linear technology y = On, where 9 isa productivity parameter. The economy's resource constraint is y =c + x, where x is net exports. The agent can borrow and lend in interna-tional capital markets at gross interest rate r, giving rise to the budgetconstraint a' = r(a + x) = r(a + On — c). The Bellman equation is

J(a) = max w(c, 1 - n) +/?(c, 1 - n)J[r(a + On - c)].c,n

The first-order and envelope conditions are

ul+piJ{a')=Ph{a')

U2+P2J(a')=fih(af)0

h(a) = fih(a')r.


The last equation tells us that in a steady state, /?(c, 1 - n)r = 1. Withconstant discounting, there is no steady state, but with more generaldiscounting schemes the form of discounting determines the steadystate and its response to changes in the environment. Here, the long-run impact of a change in (say) 9 (the wage) depends on the form of /?.Suppose f$ is a function of n only. Then the steady-state condition/?(1 — n)r = 1 determines n independently of 9\ More generally, thelong-run impact on n of a change in 9 depends on the form of the dis-count function /?(c, 1 - n). [Adapted from Epstein and Hynes (1983),Mendoza (1991), Obstfeld (1981), Schmitt-Grohe and Uribe (2002), andShi (1994).]

Example 4 (dynamically inconsistent preferences) Suppose preferences asof date t are given by:

Ut(tc) = u(ct) +dfiu(ct+1) +Sp2u(ct+2) +5p

3u(ct+3) + • • •

with 0 < 8 < 1. When 8 = 1, this reduces to the time-additive utilityfunction (2). Otherwise, we discount utility in periods f-fl ,£ + 2,f + 3 , . . . by Sj3,Sp2,SI]3,.... A little effort should convince you thatthese preferences cannot be put into stationary recursive form. In fact,they are dynamically inconsistent in the sense that preferences over(say) (Cf+i,Cf+2) at date t are different from preferences at t + 1. (Note,for example, the marginal rates of substitution between ct+\ and ct+2 att and t + 1.) This structure is ruled out by Koopmans, who begins withthe presumption of a consistent set of preferences. We'll return to thisexample in Section 7. [Adapted from Harris and Laibson (2003) andPhelps and Pollack (1968).]

3. Risk

Our next topic is risk, which we consider initially in a static setting.Our theoretical agent makes choices that have risky consequences orpayoffs and has preferences over those consequences and their proba-bilities. To be specific, let us say that the state z is drawn with probabil-ity p(z) from the finite set Jf = { l , 2 , . . . , Z } . Consequences (c, say)depend on the state. Having read Debreu's Theory of Value or the like,we might guess that with the appropriate technical conditions, theagent's preferences can be represented by a utility function of state-contingent consequences (consumption):


At this level of generality, there is no mention of probabilities, al-though we can well imagine that the probabilities of the various stateswill show up somehow in U, as they do in equation (1). In this section,we regard the probabilities as known, which you might think of asan assumption of risk or rational expectations. We consider unknownprobabilities (ambiguity) in Sections 5 and 6.

We prefer to work with a different (but equivalent) representation ofpreferences. Suppose, for the time being, that c is a scalar; very little ofthe theory depends on this, but it streamlines the presentation. We de-fine the certainty equivalent of a set of consequences as a certain conse-quence /x that gives the same level of utility:

If LI is increasing in all its arguments, we can solve this for thecertainty-equivalent function /z({c(z)}). Clearly // represents the samepreferences as U, but we find its form particularly useful. For onething, it expresses utility in payoff (consumption) units. For another, itsummarizes behavior toward risk directly: since the certainty equiva-lent of a sure thing is itself, the impact of risk is simply the differencebetween the certainty equivalent and expected consumption.

The traditional approach to preferences in this setting is expectedutility, which takes the form:

Z)}) = YP{Z)U[C{Z)]=EU{C)

or

a special case of (1). Preferences of this form are used in virtually allmacroeconomic theory, but decades of experimental research havedocumented numerous difficulties with it. Among them: people seemmore averse to bad outcomes than expected utility implies. See, for ex-ample, the summaries in Kreps (1988, Chapter 14) and Starmer (2000).We suggest the broader Chew-Dekel class of risk preferences, whichallows us to account for some of the empirical anomalies of expectedutility without giving up its analytical tractability.


3.1 The Chew-Dekel Risk Aggregator

Chew (1983, 1989) and Dekel (1986) derive a class of risk preferencesthat generalizes expected utility, yet leads to first-order conditions thatare linear in probabilities, hence easily solved and amenable to econo-metric analysis. In the Chew-Dekel class, the certainty equivalent func-tion JX for a set of payoffs and probabilities {c(z),p(z)} is definedimplicitly by a risk aggregator M satisfying

z(4)

This is Epstein and Zin's (1989) equation (3.10) with M = F + //. Chew(1983,1989) and Dekel (1986, Section 2) show that such preferences sat-isfy a weaker condition than the notorious independence axiom thatunderlies expected utility. We assume M has the following properties:(i) M(m, m) = m (sure things are their own certainty equivalents), (ii)M is increasing in its first argument (first-order stochastic domi-nance), (iii) M is concave in its first argument (risk aversion), and (iv)M(kc, km) — kM(c, m) for k > 0 (linear homogeneity). Most of the ana-lytical convenience of the Chew-Dekel class follows from the linearityof equation (4) in probabilities.

In the examples that follow, we focus our attention on the followingtractable members of the Chew-Dekel class:

• Expected utility. A version with constant relative risk aversion isimplied by

M(c,m) = c*ml~*/oi + ra(l - I/a).

If a < 1, M satisfies the conditions outlined above. Applying (4), wefind

the usual expected utility with a power utility function.

• Weighted utility. Chew (1983) suggests a relatively easy way to gener-alize expected utility given (4): weight the probabilities by a function ofoutcomes. A constant-elasticity version follows from

M(c,m) = {c/mYcW^/a + m[l - (c/m)?/a].


For M to be increasing and concave in c in a neighborhood of m, theparameters must satisfy either (a) 0 < y < 1 and a + y < 0 or (b) y < 0and 0 < a + y < 1. Note that (a) implies a < 0, (b) implies a > 0, andboth imply a + 2y < 1. The associated certainty equivalent functionsatisfies

7+oc

^ ^ x c ( z ]r- ST •n(y\rlv\

2^ix tJ\x)L\x)

where

p(Z)c(zyp(z) =

:xp(x)c(XyThis version highlights the impact of bad outcomes: they get greaterweight than with expected utility if y < 0, less weight otherwise.

• Disappointment aversion. Gul (1991) proposes another model that in-creases sensitivity to bad events (disappointments). Preferences aredefined by the risk aggregator

fcam1"a/a + m(l - I/a) c>mM(c m) = <

v ' ' \ c a m 1 - 7 a + m(l - I / a ) +S(cam1~cc - m) /a c 0. When (5 = 0, this reduces to expected utility. Other-wise, disappointment aversion places additional weight on outcomesworse than the certainty equivalent. The certainty equivalent functionsatisfies

where I(x) is an indicator function that equals 1 if x is true and 0 other-wise and

It differs from weighted utility in scaling up the probabilities of all badevents by the same factor, and scaling down the probabilities of goodevents by a complementary factor, with good and bad defined asbetter and worse, respectively, than the certainty equivalent. All threeexpressions highlight the recursive nature of the risk aggregator M: weneed to know the certainty equivalent to know which states are bad sothat we can compute the certainty equivalent (and so on).


Each of these models is described in Epstein and Zin (2001). Other trac-table preferences include semiweighted utility (Epstein and Zin, 2001),generalized disappointment aversion (Routledge and Zin, 2003), andrank-dependent preferences (Epstein and Zin, 1990). All but the lastone are members of the Chew-Dekel class.

One source of intuition about these preferences is their state-spaceindifference curves, examples of which are pictured in Figure 2. For

1 2Expected Utility

1 2Weighted Utility

1 2Disappointment Aversion

1 2All Together

Figure 2State-space indifference curves with Chew-Dekel preferences. The figure contains indif-ference curves for three members of the Chew-Dekel class of risk preferences. In eachcase, the axes are consumption in state 1 and state 2 and states are equally likely. Therisk preferences are expected utility (upper left, a. = 0.5), weighted utility (upper right,bold line, y = —0.25), and disappointment aversion (lower left, bold line, S — 0.5). Forweighted utility and disappointment aversion, expected utility is pictured with a lighterline for comparison. For disappointment aversion, the indifference curve is the upper en-velope of two indifference curves, each based on a different set of transformed probabil-ities. The extensions of these two curves are shown as dashed lines. The lower rightfigure has all three together: expected utility (dashed line), weighted utility (solid line),and disappointment aversion (dash-dotted line). Note that disappointment aversion ismore sharply convex than weighted utility near the 45-degree line (the effect of first-order risk aversion), but less convex far away from it.


the purpose of illustration, suppose there are two equally likely states(Z = 2, p(l) = p(2) = 1/2). The 45-degree line represents certainty[c(l) = c{2)\. Since preferences are linear homogeneous, the unit in-difference curve (JU = 1) completely characterizes preferences. For ex-pected utility, the unit indifference curve is

//(EU) = [0.5c(l)a + 0.5c(2)a]1/a = 1.

This is the usual convex arc with a slope of —1 (the odds ratio) at the45-degree line. As we decrease a, the arc becomes more convex. Forweighted utility, the unit indifference curve is

//(WIT) = — K ' = 1

Drawn for the same value of a and a modest negative value of y, it ismore convex than expected utility, suggesting greater risk aversion.With disappointment aversion, the equation governing the indifferencecurve depends on whether c(l) is larger or smaller than c(2). If it'ssmaller (so that z = 1 is the bad state), the indifference curve is

If it's larger, we switch the two states around. To express thismore compactly, define sets of transformed probabilities, p1 =[(l+S)/(2 + S),l/(2 + S)] (when z = 1 is the bad state) and p2 =[1/(2 + S), (1 + S)/(2 + 3)] (when z = 2 is the bad state). Then the indif-ference curve can be expressed as

We'll see something similar in Section 6. For now, note that the indif-ference curve is the upper envelope of two curves based on differentsets of probabilities. The envelope is denoted by a solid line, and theextensions of the two curves by dashed lines. The result is an indif-ference curve with a kink at the 45-degree line, where the bad stateswitches. (As we cross from below, the bad state switches from 2to 1.)

Another source of intuition is the sensitivity of certainty equivalentsto small risks. For the two-state case discussed above, consider thecertainty equivalent of the outcome c(l) — 1 — a and c(2) = 1 + a for


small a > 0, thereby defining the certainty equivalent as a function ofa. How much does a small increase in a reduce fxl For expected utility,a second-order Taylor series expansion of pi(a) around a = 0 is

This familiar bit of mathematics suggests 1 — a as a measure of riskaversion. For weighted utility, a similar approximation yields

which suggests 1 — a — 2y as a measure of risk aversion. Note thatneither expected utility nor weighted utility has a linear term: agentswith these preferences are effectively indifferent to very small risks.For disappointment aversion, however, the Taylor series expansionis

+ 8) v

The linear term tells us that disappointment aversion exhibits first-orderrisk aversion, a consequence of the kink in the indifference curve.

3.2 Examples

Example 5 (certainty equivalents for log-normal risks) We illustrate thebehavior of Chew-Dekel preferences in an environment in which theimpact of risk on utility is particularly transparent. Define the riskpremium on a risky consumption distribution by rp = log[E(c)///(c)],the logarithmic difference between consumption's expectation and itscertainty equivalent. Suppose consumption is log-normal: log c(z) =KI+KI/2Z, with z distributed N(0,1). Recall that if logx~N(a,b),then log E(x) = a + b/2 ["Ito's lemma," equation (42) of Appendix9.2]. Since log c ~ N(K\,K2), expected consumption is exp(K- + KJ/2).Similarly, the certainty equivalent for expected utility is /i =exp(/ci + (XK2/2) and the risk premium is rp = (1 — OC)K2/2. The pro-portionality factor (1 — a) is the traditional coefficient of relative riskaversion. Weighted utility is not quite kosher in this context (M is con-cave only in the neighborhood of ju), but the example neverthelessgives us a sense of its properties. Using similar methods, we find thatthe certainty equivalent is fi — exp(/ci + (a + 2y)K2/2) and the risk


0.25

0.2

0.15

0.1

0.05

n

-

-

- Disappointment Aversion

x '

• yy

••yy Weighted Utility

„ - " Expected Utility

-

0.1 0.2 0.3 0.4Variance of Log Consumption

0.5 0.6

Figure 3Risk and risk premiums with Chew-Dekel preferences. The figure illustrates the relationbetween risk and risk premiums discussed in Example 5 for three members of the Chew-Dekel class of risk preferences. The preferences are: expected utility (dashed line),weighted utility (solid line), and disappointment aversion (dash-dotted line). The pointis the nonlinearity of disappointment aversion: the ratio of the risk premium to risk isgreater for small risks than large ones. Parameter values are the same as Figure 2.

premium is rp = (1 — a — 2y)/C2/2. Note that the risk premium is thesame as expected utility with parameter a' = a + 2y. This equivalenceof expected utility and weighted utility doesn't extend to other distri-butions, but it suggests that we might find some difficulty distinguish-ing between the two in practice. For disappointment aversion, we findthe certainty equivalent using mathematics much like that underlyingthe Black-Scholes formula:

jUa _ eocKi+a

2K2/2 , ^

V.1/2

where


(a = S = 0.5, y = —0.25). As you might expect, disappointment aver-sion implies proportionately greater aversion to small risks than largeones; in this respect, it is qualitatively different from expected utilityand weighted utility. Routledge and Zin's (2003) generalized dis-appointment aversion does the reverse: it generates greater aversionto large risks. Different sensitivity to large and small risks providesa possible method to distinguish such preferences from expectedutility.

Example 6 (portfolio choice with Chew-Dekel preferences) One strength ofthe Chew-Dekel class is that it leads to first-order conditions thatare easily solved and used in econometric work. Consider an agentwith initial net assets «o who invests fractions roina risky asset with(gross) return r(z) in state z and 1 — w in a risk-free asset with returnYQ. For an arbitrary choice of w, consumption in state z is c(z) =ao[ro + w(r{z) — r0)}. The portfolio choice problem might then bewritten as

max ju[ao{ro + w(r{z) - ro)}\ = a0 max ju[rQ + w(r(z) - r0)],

the second equality stemming from the linear homogeneity of ju. Thedirect approach to this problem is to choose w to maximize ju, and insome cases we'll do that. For the general Chew-Dekel class, however,we may not have an explicit expression for the certainty equivalentfunction. In those cases, we use equation (4):

max /u[{r0 + w(r(z) - r0)}] = max ^ p{z)M[r0 + w(r(z) - r0),//*],2

where /n* is the maximized value of the certainty equivalent function.The problem on the righthand side has first-order condition

p(z)Mi[r0 + w{r{z) - ro),,u*][r(z) - r0]

= ElM^r, + w(r - ro),f){r - r0)] = 0. (5)

(There are M2 terms, too, but you might verify for yourself that theycan be eliminated.) We find the optimal portfolio by solving the first-order condition and (4) simultaneously for w and //*. The same con-ditions can also be used in econometric work to estimate preferenceparameters.


To see how you might use (5) to determine w, consider a numeri-cal example with two equally likely states and returns YQ = 1.01,r(l) = 0.90, and r(2) = 1.24 (the "equity premium" is 6%). With ex-pected utility, the first-order condition is

(^*)a"1(l -fi)Y, P(z)(ro + w[r{z) - rc])""1^) - r0] = 0.2

Note that pi* drops out and we can solve for w independently. Fora = 0.5, the solution is w — 4.791, which implies pi* = 1.154. The resultis the dual of the equity premium puzzle: with modest risk aversion,the observed equity premium induces a huge long position in the riskyasset, financed by borrowing. With disappointment aversion, the first-order condition is

+ p(2)(r0 + w[r(2) - ro])^1[r(2) - r0] = 0,

since z = 1 is the bad state. For 8 = 0.5, w = 2.147 and pi* = 1.037.[Adapted from Epstein and Zin (1989, 2001).]

Example 7 (portfolio choice with rank-dependent preferences) Rank-dependent preferences are an interesting alternative to the Chew-Dekel class. We rank states so that the payoffs c(z) are increasing in zand define the certainty equivalent function by

where g is an increasing function satisfying g{0) = 0 and g(l) = 1,P(z) = J2l=i P(u) *s t r i e cumulative distribution function, and p[z) —g\P{z)\ — g[P(z — 1)] is a transformed probability. If g(p) = p, this issimply expected utility. If g is concave, these preferences exhibit riskaversion even if u is linear. However, since pi is nonlinear in probabil-ities, it cannot be expressed in Chew-Dekel form. At the end of thissection, we discuss the difficulties this raises for econometric estima-tion. In the portfolio choice problem, the first-order condition is

] T p{z)ui [c{z)\ [r(z) - r0] = 0, (6)

which is readily solved if we know the probabilities. [Adapted fromEpstein and Zin (1990) and Yaari (1987).]


Example 8 (risk sharing) Consider a Pareto problem with two agentswho divide a given risky aggregate endowment y(z). If their certaintyequivalent functions are identical and homogeneous of degree 1, eachagent consumes the same fraction of the aggregate endowment in allstates. The problem is more interesting if the agents have differentpreferences. Let us say that two agents, indexed by i, have certaintyequivalent functions nl[cl(z)}. A Pareto optimal allocation solves:choose {c1(z), c2(z)} to maximize /il subject to c1(z) + c2(z) < y(z) andju2 > p, for some number Jx. If X is the Lagrange multiplier on the sec-ond constraint, the first-order conditions have the form

dju1 _ d/u2

dcl{z) dc2{z)'

With Chew-Dekel risk preferences, the derivatives have the form:

= p{z)Mi[c\z),Mi}/(l-Yv(x)Mi[ci(x),Mi]X

This expression is not particularly user-friendly, but in principle wecan solve it numerically for specific functional forms. With expected(power) utility, an optimal allocation solves

which implies allocation rules that we can express in the form cl —sl(y)y. If we substitute into the optimality condition and differentiate,we find ds1/dy > 0 if ai > 0C2: the less risk averse agent absorbs a dis-proportionate share of the risk.

3.3 Discussion: Moment Conditions for Preference Parameters

One of the most useful features of Chew-Dekel preferences is how eas-ily they can be used in econometric work. Since the risk aggregator (4)is linear in probabilities, we can apply method of moments estimatorsdirectly to first-order conditions.

In a typical method of moments estimator, a vector-valued function/ of data x and a vector of parameters 6 of equal dimension satisfiesthe moment conditions


E/(*,0o)=O, (7)

where 6 = 0Q is the parameter vector that generated the data. Amethod of moments estimator Oj for a sample of size T replaces thepopulation mean with the sample mean:

Under reasonably general conditions, a law of large numbers impliesthat the sample mean converges to the population mean and 6j con-verges to #o- When the environment permits a central limit theorem,we can also derive an asymptotic normal distribution for Qj. If thenumber of moment conditions (the dimension of / ) is greater than thenumber of parameters (the dimension of 6), we can apply a general-ized method of moments estimator with similar properties (see Han-sen, 1982.)

The portfolio-choice problem with Chew-Dekel preferences has ex-actly this form if the number of preference parameters is no greaterthan the number of risky assets. For each risky asset i, there is a mo-ment condition,

fi(x,O)=M1{c,fi*){ri-ro),

analogous to equation (5). In the static case, we also need to estimate/i*, which we do using equation (4) as an additional moment condition.[In a dynamic setting, a homothetic time aggregator allows us to re-place fi* with a function of consumption growth; see equation (13).]

Outside the Chew-Dekel class, estimation is a more complex activ-ity. First-order conditions are no longer linear in probabilities and donot lead to moment conditions in the form of equation (7). To estimate,say, equation (6) for rank-dependent preferences, we need a differentestimation strategy. One possibility is a simulated method of momentsestimator, which involves something like the following: (i) conjecture aprobability distribution and parameter values; (ii) given these values,solve the portfolio problem for decision rules; (iii) calculate (perhapsthrough simulation) moments of the decision rule and compare themto moments observed in the data; (iv) if the two sets of moments aresufficiently close, stop; otherwise, modify parameter values and returnto step (i). All of this can be done, but it highlights the econometricconvenience of Chew-Dekel risk preferences.


4. Time and Risk

We are now in a position to describe nonadditive preferences in adynamic stochastic environment like that illustrated by Figure 1. Youmight guess that the process of specifying preferences over time andstates of nature is simply a combination of the two. In fact, the combi-nation raises additional issues that are not readily apparent. We touchon some of them here; others come up in the next two section!?.

4.2 Recursive Preferences

Consider the structure of preferences in a dynamic stochastic envi-ronment. In the tradition of Kreps and Porteus (1978), Johnsen andDonaldson (1985), and Epstein and Zin (1989), we represent a class ofrecursive preferences by

Ut = V[ut,Mt{UM)]i (8)

where Ut is shorthand for utility starting at some date-t history zf, Ut+i

refers to utilities for histories z m = (zf,zf+i) stemming from zl, ut is

date-t utility, V is a time aggregator, and jut is a certainty-equivalentfunction based on the conditional probabilities p{zt+\\z

i). This struc-ture is suggested by Kreps and Porteus (1978) for expected utility cer-tainty equivalent functions. Epstein and Zin (1989) extend their workto stationary infinite-horizon settings and propose the more generalChew-Dekel class of risk preferences. As in Section 2, such preferencesare dynamically consistent, history independent, future independent,and stationary. They are also conditionally independent in the sense ofJohnsen and Donaldson (1985): preferences over choices at any historyat date t (zf, for example) do not depend on other histories that mayhave (but did not) occur {zt # zf). You can see this in Figure 1: if weare now at the node marked (A), then preferences do not depend onconsumption at nodes stemming from (B) denoting histories that canno longer occur.

If equation (8) seems obvious, think again. If you hadn't read theprevious paragraph or its sources, you might just as easily propose

another seemingly natural combination of time and risk preference.This combination, however, has a serious flaw: it implies dynamicallyinconsistent preferences unless it reduces to equation (1). See Kreps


and Porteus (1978) and Epstein and Zin (1989, Section 4). File away forlater the idea that the combination of time and risk preference can raisesubtle dynamic consistency issues.

We refer to the combination of the recursive structure (8) and anexpected utility certainty equivalent as Kreps-Porteus preferences.A popular parametric example consists of the constant elasticityaggregator,

1/p, . (9)

and the power certainty equivalent,

with />, a < 1. Equations (9) and (10) are homogeneous of degree 1 withconstant discount factor /?. This is more restrictive than the aggregatorswe considered in Section 2, but linear homogeneity rules out more gen-eral discounting schemes: it implies that indifference curves have thesame slope along any ray from the origin, so their slope along the 45-degree line must be the same, too. If U is constant, the weights (1 — fi)and P define U — u as the (steady-state) level of utility. It is common torefer to 1 — a as the coefficient of relative risk aversion and 1/(1 — p) asthe intertemporal elasticity of substitution. If p = a, the model is equiv-alent to one satisfying equation (1), and intertemporal substitution isthe inverse of risk aversion. More generally, the Kreps-Porteus struc-ture allows us to specify risk aversion and intertemporal substitutionindependently. Further, a Kreps-Porteus agent prefers early resolutionof risk if a < p; see Epstein and Zin (1989, Section 4). This separation ofrisk aversion and intertemporal substitution has proved to be not onlya useful empirical generalization but an important source of intuitionabout the properties of dynamic models.

We can generate further flexibility by combining (8) with a Chew-Dekel risk aggregator (4), thereby introducing Chew-Dekel risk pref-erences to dynamic environments. We refer to this combination asEpstein-Zin preferences.

4.2 Examples

Example 9 (Weil's model of precautionary saving) We say thatconsumption-saving models generate precautionary saving if risk de-creases consumption as a function of current assets. In the canonicalconsumption problem with additive preferences, income risk has this


effect if the period utility function u has constant k = MIIIMI/(«H) > 0.See, for example, Ljungqvist and Sargent (2000, pp. 390-393). Bothpower utility and exponential utility satisfy this condition. With powerutility [u(c) = ca/oc], k = (a — 2)(a — 1), which is positive for a < 1 andtherefore implies precautionary saving. (In the next section, we lookat quadratic utility, which effectively sets a = 2, implying k = 0 andno precautionary saving.) Similarly, with exponential utility [u(c) ——exp(—ac)], k = 1 > 0. With Kreps-Porteus preferences, we can addressa somewhat different question: does precautionary saving depend onintertemporal substitution, risk aversion, or both? To answer this ques-tion, consider the problem characterized by the Bellman equation

J(a) = m a x { ( l - / ? K

subject to the budget constraint a' = r(a — c) + y', where p{x) =—a"1 log E exp(—ax) and {yt} ~ NID(/ci,/C2). The exponential cer-tainty equivalent p is not homogeneous of degree 1, but it is ana-lytically convenient for problems with additive risk. The parameterssatisfy p < 1, a > 0, r > 1, and 1ff

1/(1"/))^(1^ < 1. Of particular interestare p, which governs intertemporal substitution, and a, which governsrisk aversion.

The value function in this example is linear with parameters thatcan be determined by the time-honored guess-and-verify method. Weguess (we've seen this problem before) J(a) — A + Ba for parameters(A, B) to be determined. The certainty equivalent of future utility is

fi[J(a')] = fi[A + Br(a - c) + By'} =A + Br{a - c) + BKX - OCB2K2/2, (11)

which follows from equation (42) of Appendix 9.2. The first-order andenvelope conditions are

which imply

M = {L3r)1/{l-p)J(a) = {fir)1/{1-p)(A + Ba)

c = [{l-p)/B]ll{l-p)]{a) = [(1 1/

The latter tells us that the decision rule is linear, too. If we substituteboth equations into (11), we find that the parameters of the value func-tion must be:


-BaK2/2)B

They imply the decision rule

r — C\ — R}-IQ~P)rPlQ~P)\(n -I- (r — "H^IVi — Tirvrr^/?]\

The last term is the impact of risk. Apparently a necessary conditionfor precautionary saving is a > 0, so the parameter controlling precau-tionary saving is risk aversion. [Adapted from Weil (1993).]

Example 10 (Merton-Samuelson portfolio model) Our next example illus-trates the relation between consumption and portfolio decisions in iidenvironments. The model is similar to the previous example, and weuse it to address a similar issue: the impact of asset return risk on con-sumption. At each date t, a theoretical agent faces the following budgetconstraint:

at+i = (at -

where wu is the share of post-consumption wealth invested in asset iand Tit+i is its return. Returns {ru+\} are iid over time. Preferences arecharacterized by the constant elasticity time aggregator (9) and an arbi-trary linearly homogeneous certainty equivalent function. The Bellmanequation is

7(a)=max{(l-/?)c'+M/(«T}VP

subject to

a' = {a-

and J2i wi ~ 1/ where rv is the portfolio return. Since the time and riskaggregators are linear homogeneous, so is the value function, and theproblem decomposes into separate portfolio and consumption prob-lems. The portfolio problem is:

max MJO')] = {a-c) max ju[J{r')}.


Since returns are iid, the portfolio problem is the same at all dates andcan be solved using methods outlined in the previous section. Given asolution JU* to the portfolio problem, the consumption problem is:

]{a) = max{(l - p)c" + fi[(a - c K ] ' } 1 7 ' -

The first-order condition implies the decision rule c= [A/(1+A)]a,where

A = [(l-P)/p]ll{l-p\n*)-pl{l-p).

The impact of risk is mediated by JU* and involves the familiar balanceof income and substitution effects. If p < 0, the intertemporal elasticityof substitution is less than 1 and smaller /z* (larger risk premium) isassociated with lower consumption (the income effect). If p > 0, theopposite happens. In contrast to the previous example, the governingparameter is p; the impact of risk parameters is imbedded in JU*. Note,too, that the impact on consumption of a change in pi* can generally beoffset by a change in /? that leaves A unchanged. This leads to an iden-tification issue that we discuss at greater length in the next example.Farmer and Gertler use a similar result to motivate setting a = 1 (riskneutrality) in the Kreps-Porteus preference model, which leads to lin-ear decision rules even with risk to income, asset returns, and lengthof life. [Adapted from Epstein and Zin (1989), Farmer (1990), Gertler(1999), and Weil (1990).]

Example 11 (asset pricing) The central example of this section is anexploration of time and risk preference in the traditional exchangeeconomy of asset pricing. Preferences are governed by the constantelasticity time aggregator (9) and the Chew-Dekel risk aggregator (4).We characterize asset returns for general recursive preferences anddiscuss the identification of time and risk preference parameters. Webreak the argument into a series of steps.

Step (i) (consumption and portfolio choice). Consider a stationaryMarkov environment with states z and conditional probabilities p(z'\z).A dynamic consumption/portfolio problem for this environment ischaracterized by the Bellman equation

J(a,z) = max{(l - p)

subject to the budget constraint a' = (a — c) ^^ir^z^z') == (a — c) •J2j W\r[ = (a — c)r' where rp is the portfolio return. The budget con-


straint and linear homogeneity of the time and risk aggregators implylinear homogeneity of the value function: }(a, z) = ah{z) for some scaledvalue function L. The scaled Bellman equation is

L(z) = max{(l - fi)b" +/?(1 - b)p»[L(z')rp(z,z')]p}1/p,

b,w

where b = c/a. Note that L(z) is the marginal utility of wealth in state z.As in the previous example, the problem divides into separate port-

folio and consumption decisions. The portfolio decision solves: choose{wi} to maximize /u[L(z')rp(z,z')]. The mechanics are similar to Exam-ple 6. The portfolio first-order conditions are

^p(z/|z)M1[L(z/)rp(z,z

/)^]L(z/)[rI-(z,z/) -rfaz')] = 0 (12)

z'

for any two assets i and ;. Given a maximized [i, the consumption deci-sion solves: choose b to maximize L. The intertemporal first-order con-dition is

{l-p)bp-1 =p{l-b)p-lfip. (13)

If we solve for // and substitute into the (scaled) Bellman equation, wefind

The first-order condition (13) and the value function (14) allow usto express the relation between consumption and returns in almostfamiliar form. Since pi is linear homogeneous, the first-order conditionimplies ju(x'r!) — 1 for

The last equality follows from (c'/c) — (b'/b)(l — b)r' a consequenceof the budget constraint and the definition of b. The intertemporalfirst-order condition can therefore be expressed as

^x'r'v)=^WIc)p-\]llp) = l, (15)

a generalization of the tangency condition for an optimum (set themarginal rate of substitution equal to the price ratio). Similar logicleads us to express the portfolio first-order conditions (12) as

E[M1(x'r'l)xXr>-r')}=0.


If we multiply by the portfolio weight Wj and sum over /, we find

EiM^x'r'^x'rl] = E^ix'^l)^}. (16)

Euler's theorem for homogeneous functions allows us to express theright side as

Whether this is helpful depends on M. [Adapted from Epstein and Zin(1989).]

Step (ii) (equilibrium). Now shift focus to an exchange economy inwhich output growth follows a stationary Markov process: g' =y'/y = g(z'). In equilibrium, consumption equals output and the opti-mal portfolio is a claim to the stream of future output. We denote theprice of this claim by q and the price-output ratio by Q = q/y. Its returnis therefore

r'P = (q1 + y')/q = (QV + y')/(Qy) = g'(Q' + i)/Q. (17)

With linear homogeneous preferences, the equilibrium price-outputratio is a stationary function of the current state, Q(z). Asset pric-ing then consists of these steps: (a) substitute (17) into (15) and solveforQ:

(b) compute the portfolio return rp from equation (17); and (c) use (16)to derive returns on other assets.

Step (iii) (the iid case). If the economy is iid, we cannot generallyidentify separate time and risk parameters. Time and risk parametersare intertwined in (16), but suppose we were somehow able to esti-mate the risk parameters. How might we estimate the time prefer-ence parameters /? and p from observations of rp (returns) and b (theconsumption-wealth ratio)? Formally, equations (13) and (14) implythe intertemporal optimality condition

If rv is iid, p. and b are constant. With no variation in p or b, the optimal-ity condition cannot tell us both p and /?: for any value of p, we can sat-isfy the condition by adjusting the discount factor fi. The only limit tothis is the restriction /? < 1. Evidently a necessary condition for identi-fying separate time and risk parameters is that risk varies over time.


The issue doesn't arise with additive preferences, which tie time prefer-ence to risk preference. [Adapted from Kocherlakota (1990) and Wang(1993).]

Step (iv) (extensions). With Kreps-Porteus preferences and non-iid returns, the model does somewhat better in accounting for assetreturns. Nevertheless, it fails to provide an entirely persuasive accountof observed relations between asset returns and aggregate consump-tion. Roughly speaking, the same holds for more general risk prefer-ence specifications, although the combination of exotic preferencesand time-varying risk shows promise. [See Bansal and Yaron (2004);Epstein and Zin (1991); Lettau, Ludvigson, and Wachter (2003); Rout-ledge and Zin (2003); Tallarini (2000); and Weil (1989).]

Example 12 (risk sharing) With additive preferences and equal dis-count factors, Pareto problems generate constant weights on agents'utilities over time and across states of nature, even if period/stateutility functions differ. With Kreps-Porteus preferences, differences inrisk aversion lead to systematic drift in the weights. To be concrete,suppose states z follow a Markov chain with conditional probabilitiesp(z'\z). Aggregate output is y{z). Agents have the same aggregator,V(c,/i) = (cp + /3jup)/p, but different certainty equivalent functions,

for state-dependent utility x. The Bellman equation for the Pareto prob-lem is

J ( , )c,{wz,}

subject to

Here, c and wz> refer to consumption and promised future utility of thesecond agent. The first-order and envelope conditions imply

h(w,z) = -


The first equation leads to the familiar allocation rule c =[1 + kl'^p~l^]~ly(z). If a.\ ̂ CC2, the weight X will generally vary overtime. [Adapted from Anderson (2004) and Kan (1995).]

Example 13 (habits, disappointment aversion, and conditional indepen-dence) Habits and disappointment aversion both assess utility bycomparing consumption to a benchmark. With disappointment aver-sion, the benchmark is the certainty equivalent. With habits, thebenchmark is a function of past consumption. Despite this apparentsimilarity, there are a number of differences between them. One is tim-ing: the habit is known and fixed when current decisions are made,while the certainty equivalent generally depends on those decisions.Another is that disappointment aversion places restrictions on thebenchmark that have no obvious analog in the habit model. A third isthat habits take us outside the narrowly defined class of recursive pref-erences summarized by equation (8): they violate the assumption ofconditional independence. Why? Because preferences at any node inthe event tree depend on past consumption through the habit, whichin turn depends on nodes that can no longer be reached. In Figure 1,for example, decisions at node (A) depend on the habit, which waschosen at (say) the initial node ZQ and therefore depends on anythingthat could have happened from there on, including (B) and its succes-sors. The solution, of course, is to define preferences conditional on ahabit state variable and proceed in the natural way.

4.3 Discussion: Distinguishing Time and Risk Preference

The defining feature of this class of preferences is the separation oftime preference (summarized by the aggregator V) and risk preference(summarized by the certainty equivalent function /J). In the functionalforms used in this section, time preference is characterized by a dis-count factor and an intertemporal substitution parameter. Risk prefer-ence is characterized by risk aversion and possibly other parametersindicated by the Chew-Dekel risk aggregator. Therefore, we haveadded one or more parameters to the conventional additive utilityfunction (1). Examples suggest that the additional parameters may behelpful in explaining precautionary saving, asset returns, and the inter-temporal allocation of risk.

A critical question in applications is whether these additional param-eters can be identified and estimated from a single time series real-


ization of all the relevant variables. If so, we can use the methods out-lined in the previous section: apply a method of moments estimator tothe first-order conditions of the problem of interest. Identificationhinges on the nature of risk. If risk is iid, we cannot identify separatetime and risk parameters. This is clear in examples, but the logic isboth straightforward and general: we need variation over time to iden-tify time preference. A more formal statement is given by Wang (1993).

5. Risk-Sensitive and Robust Control

Risk-sensitive and robust control emerged in the engineering literaturein the 1970s and were brought to economics and developed further byHansen and Sargent, their many coauthors, and others. The most pop-ular version of risk-sensitive control is based on Kreps-Porteus prefer-ences with an exponential certainty equivalent function. Robust controlconsiders a new issue: decision making when the agent does not knowthe probability model generating the data. The agent considers insteada range of models and makes decisions that maximize utility given theworst possible model. The same issue is addressed from a differentperspective in the next section. Much of this work deals with linear-quadratic-guassian (LQG) problems, but the ideas are applicable moregenerally. We start by describing risk-sensitive and robust control ina static scalar LQG setting, where the insights are less cluttered by al-gebra. We go on to consider dynamic LQG problems, robust controlproblems outside the LQG universe, and challenges of estimating—and distinguishing between—models based on risk-sensitive and ro-bust control.

5.1 Static Control

Many of the ideas behind risk-sensitive and robust control can beillustrated with a static, scalar example. We consider traditional opti-mal control, risk-sensitive control, and robust control as variants of thesame underlying problem. The striking result is the equivalence of op-timal decisions made under risk-sensitive and robust control.

In our example, an agent maximizes some variant of a quadratic re-turn function,

subject to the linear constraint,


Bv + C(w + s), (18)

where v is a control variable chosen by the agent, x is a state vari-able that is controlled indirectly through v, XQ is a fixed initial value,(Q,R) > 0 are preference parameters, (A,B,C) are nonzero parametersdescribing the determination of x, s ~ N(0,1) is noise, and w is a dis-tortion of the model that we'll describe in greater detail when we getto robust control. The problem sets up a trade-off between the cost(Qv2) and potential benefit (Rx2) of nonzero values of v. If you've seenLQG control problems before, most of this should look familiar.

Optimal control. In this problem and the next one, we set w = 0,thereby ruling out distortions. The control problem is: choose v to max-imize Eu given the constraint (18). Since

Eu = -[Qv2 + R{Ax0 + Bv)2} - RC2, (19)

the objective functions with and without noise differ only by a con-stant. Noise therefore has no impact on the optimal choice of v. Forboth problems, the optimal v is

v = -(Q + B2R)-\ABR)x0.

This solution serves as a basis of comparison for the next two.Risk-sensitive control. Continuing with zv — 0, we consider an alterna-

tive approach that brings risk into the problem in a meaningful way:we maximize an exponential certainty equivalent of u,

ju(u) = -a" 1 log E exp(-au),

where a > 0 is a risk aversion parameter. (This is more natural in adynamic setting, where we would compute the certainty equivalent offuture utility a la Kreps and Porteus.) We find ju(u) by applying for-mula (43) of Appendix 9.2:

fi(u) = -(1/2) log(l -

- [Qv2 + [R/(l - 2aRC2)](Ax0 + Bv)2} (20)

as long as 1 - 2aRC2 > 0. This condition places an upper bound on therisk aversion parameter a. Without it, the agent can be so sensitive torisk that her objective function is negative infinity regardless of thecontrol. The first term on the right side of (20) does not depend on v orx, so it has no effect on the choice of v. The important difference from(19) is the last term: the coefficient of (AXQ + Bv) is larger than R, mak-


ing the agent more willing to tolerate nonzero values of v to bring xclose to zero. The optimal v is

v = - (Q + B2R - aQRC2)~\ABR)x0.

If a = 0 (risk neutrality) or C = 0 (no noise), this is the same as the opti-mal control solution. If a > 0 and C ^ 0, the optimal choice of v islarger in absolute value because risk aversion increases the benefit ofdriving x to zero.

Robust control. Our third approach is conceptually different. Webring back the distortion w and tell the following story: We are playinga game against a malevolent nature, who chooses w to minimize ourobjective function. If our objective were to maximize Eu, then w wouldbe infinite and our objective function would be minus infinity regard-less of what we do. Therefore, let us add a penalty (to nature) of 6w2,making our objective function

min Eu + 8w2.w

The parameter 0 > 0 has the effect of limiting how much naturedistorts the model, with small values of 0 implying weaker limits onnature. The minimization implies

+ Bv),

making the robust control objective function

min Eu + Qw2 = -\Qv2 + [R/(l - e~lRC2)}{AxQ + Bv)2] - RC2. (21)

The remarkable result: if we set 9~ = 2a, the robust control objectivediffers from the risk-sensitive control objective (20) only by a constant,so it leads to the same choice of v. As in risk-sensitive control, thechoice of v is larger in absolute value, in this case to offset the impactof w. There is, once again, a limit on the parameter: where a wasbounded above, 9 is bounded below. An infinite value of 6 reproducesthe optimal control objective function and solution.

An additional result applies to the example: risk-sensitive and robustcontrol are observationally equivalent to the traditional control prob-lem with suitably adjusted R. That is, if we replace R in equation (19)with

R = R/(l - 2aRC2) =R + 2aR2C2/(l - 2aRC2) > R, (22)


then the optimal control problem is equivalent to risk-sensitive control,which we've seen is equivalent to robust control. If Q and R are func-tions of more basic parameters, it may not be possible to adjust Rin this way, but the exercise points to the qualitative impact on thecontrol: be more aggressive. This result need not survive beyond thescalar case, but it's suggestive.

Although risk-sensitive and robust control lead to the same decision,they are based on different preferences and give the decision differentinterpretations. With risk-sensitive control, we are concerned with riskfor traditional reasons, and the parameter a measures risk aversion.With robust control, we are concerned with model uncertainty (possi-ble nonzero values of w). To deal with it, we make decisions that max-imize given the worst possible specification error. The parameter 9controls how bad the error can be.

Entropy constraints. One of the most interesting developments inrobust control is a procedure for setting 9: namely, choose 9 to limitthe magnitude of model specification error, with specification errormeasured by entropy. We define the entropy of transformed probabil-ities p relative to reference probabilities p by

KP\ V) = E ViA log[p(z)/p(z)] = E log(p/p), (23)2

where the expectation is understood to be based on p. Note that I(p; p)is nonnegative and equals zero when p = p. Since the likelihood isthe probability density function expressed as a function of parameters,entropy can be viewed as the expected difference in log-likelihoodsbetween the reference and transformed models, with the expectationbased on the latter.

In a robust control problem, we can limit the amount of specificationerror faced by an agent by imposing an upper bound on I: consider(say) only transformations p such that I(p; p) < IQ for some positivenumber 7o. This entropy constraint takes a particularly convenient formin the normal case. Let p be the density of x implied by equation (18)and p the density with w = 0:

p(x) = (2nC2)~1/2 exp[-(x - Ax0 - Bv - Cw)2/2C2}

= {2nC2)-l/2 exp[-e2/2]

p(x) = (2nC2y1/2 exp[-(j - Ax0 - Bv)2/2C2}

= (2TTC2)~1/2 exp[-(w + e)2/2].


Relative entropy is

I{p; p) = E(w2/2 + WE) = w2j2.

If we add the constraint w2/2 < Jo to the optimal control objective (19),the new objective is

min -[Qv2 + R(Ax0 + Bv + Cw)2] - RC2 + 6(w2 - 2I0),

where 6 is the Lagrange multiplier on the constraint. The only differ-ence from the robust control problem we discussed earlier is that 9 isdetermined by IQ. LOW values of IQ (tighter constraints) are associatedwith high values of 6, so the lower bound on 6 is associated with anupper bound on Io.

Example 14 (Kydland and Prescott's inflation game) A popular macro-economic policy game goes like this: the government chooses inflationq to maximize the quadratic return function,

subject to the Phillips curve,

y=yo + B(q- qe) + C(w + e),

where y is the deviation of output from its social optimum, qe is ex-pected inflation, (R,B,C) are positive parameters, t/o is the noninfla-tionary level of output, and s ~ N(0,1). We assume i/o < 0, whichimparts an inflationary bias to the economy.

This problem is similar to our example, with one twist: we assumeqe is chosen by private agents to equal the value of q they expect thegovernment to choose (another definition of rational expectations) buttaken as given by the government (and nature). Agents know themodel, so they end up setting qe = q. A robust control version of thisproblem leads to the optimization:

max min -E(q2 + R[y0 + B(p - pe) + C(w + e)]2) + 9w2.

Cj W

Note that we can do the min and max in any order (the min-max theo-rem). We do both at the same time, which generates the first-order con-ditions

q + RB[y0 + B(q - qe) + Cw] = 0

-Ow + RC[y0 + B(q - qe) + Cw] = 0.


Applying the rational expectations condition qe = q leads to

RB \ ( 9~lRC \i, w = \ z i/n.

Take 9~l = 0 as the benchmark. Then q = -RBy0 > 0 (the inflationarybias we mentioned earlier) and w = 0 (no distortions). For smallervalues of 9 > RC2, inflation is higher. Why? Because negative valuesof w effectively lower the noninflationary level of output (it becomest/o + Cw), leading the government to tolerate more inflation. As 9approaches its lower bound of -RC2, inflation approaches infinity. If wetreat this as a constraint problem with entropy bound w1 /2 < Jo, thenw = —(2Jo) (recall that w < 0) and the Lagrange multiplier 6 is re-lated to Jo by

e = RC2-RCy0/(2I0)1/2.

The lower bound on 6 corresponds to an upper bound on Jo. Allof this is predicated on private agents understanding the govern-ment's decision problem, including the value of 9. [Adapted fromHansen and Sargent (2004, Chapter 5) and Kydland and Prescott(1977).]

Example 15 (entropy with three states) With three states, the constraintl{p\ p) < h is two-dimensional since the probability of the third statecan be computed from the other two. Figure 4 illustrates the con-straint for the reference probabilities p{l) = p(2) = p(3) = 1/3 (thepoint marked +) and Jo =0.1. The boundary of the constraint set isthe egg shape. By varying Jo, we vary the size of the constraint set.Chew-Dekel preferences can be viewed from the same perspective.Disappointment aversion, for example, is a one-dimensional class ofdistortions. If the first state is the only one worse than the cer-tainty equivalent, the transformed probabilities are p{\) — (1 +S)p(l)/[1 +Sp(l)}, p{2) = p(2)/[l + 0. By varying S subject to the con-straint I(d) < Jo, we produce the line shown in the figure. (It hits theboundary at S — 1.5.) The interpretation of disappointment aversion,however, is different: in the theory of Section 3, the line represents dif-ferent preferences, not model uncertainty.


0.1 0.3 0.4 0.5 0.6Probability of State 1

0.7 0.8 0.9

Figure 4Transformed probabilities: Entropy and disappointment aversion. The figure illustratestwo sets of transformed probabilities described in Example 15: one set generated by anentropy constraint and the other by disappointment aversion. The bold triangle is thethree-state probability simplex. The "+" in the middle represents the reference probabil-ities: p(l) = p(2) = p(3) = 1/3. The area inside the egg-shaped contour represents trans-formed probabilities with entropy less than 0.1. The dashed line represents probabilitiesimplied by disappointment aversion with 5 between 0 to 1.5.

5.2 Dynamic Control

Similar issues and equations arise in dynamic settings. The tradi-tional linear-quadratic control problem starts with the quadratic returnfunction,

u(vt,Xt) = —(vjQvt + xjRxt + 2xJSvt),

where v is the control and x is the state. Both are vectors, and (Q, R, S)are matrices of suitable dimension. The state evolves according to thelaw of motion

xt+i = Axt + Bvt + C(wt + em), (24)


where w is a distortion (zero in some applications) and {et} ~ NID(0,1)is random noise. We use these inputs to describe optimal, risk-sensitive, and robust control problems. As in the static example, thecentral result is the equivalence of decisions made under risk-sensitiveand robust control. We skip quickly over the more torturous algebraicsteps, which are available in the sources listed in Appendix 9.1.

Optimal control. We maximize the objective function:

t=o

subject to (24) and wt = 0. From long experience, we know that thevalue function takes the form

J(x) = -xTPx - q (25)

for a positive semidefinite symmetric matrix P and a scalar q. The Bell-man equation is

-xTPx -q = max{-(vTQv + xTRx + 2xTSv)V

- 0E[(Ax + Bv + Ce')TP(Ax + Bv + Ce') +q}}. (26)

Solving the maximization in (26) leads to the Riccati equation

P = R + /3ATPA - ((SAJPB + S)(Q + pBJPByl(pBJPA + ST). (27)

Given a solution for P, the optimal control is v = —Fx, where

T 1 T A + ST). (28)

As in the static scalar case, risk is irrelevant: the control rule (28) doesnot depend on C. You can solve such problems numerically by iter-ating on the Riccati equation: make an initial guess of P (we use I),plug it into the right side of (27) to generate the next estimate ofP, and repeat until successive values are sufficiently close together.See Anderson, Hansen, McGrattan, and Sargent (1996) for algebraicdetails, conditions guaranteeing convergence, and superior computa-tional methods (the doubling algorithm, for example).

Risk-sensitive control. Risk-sensitive control arose independently butcan be regarded as an application of Kreps-Porteus preferences usingan exponential certainty equivalent. The exponential certainty equiva-lent introduces risk into decisions without destroying the quadraticstructure of the value function. The Bellman equation is


JO) = max{u(v,x)

where the maximization is subject to x ' = Ax + Bv + Ce' and //(/) =—a"1 log E exp(—a/). If the value function has the quadratic form of(25), the multivariate analog to (43) gives us

ju[J(Ax + Bv + Ce')] = - (1 /2) log|J - 2aCTPC|

where

P = P + 2aPC(I - 2zCTPCylCTP (29)

as long as \I — 2aCTPC| > 0. Each of these pieces has a counterpart inthe static case. The inequality again places an upper bound on the riskaversion parameter a; for larger values, the integral implied by the ex-pectation diverges. Equation (29) corresponds to (22); in both equa-tions, risk sensitivity increases the agent's aversion to nonzero valuesof the state variable. Substituting P into the Bellman equation and max-imizing leads to a variant of the Riccati equation,

P = R + pATPA - {PAJPB + S)(Q + PBTPB)-\pBJPA + ST), (30)

and associated control matrix,

A direct (if inefficient) solution technique is to iterate on equations (29)and (30) simultaneously. We describe another method shortly.

Robust control. As in our static example, the idea behind robust con-trol is that a malevolent nature chooses distortions w that reduce ourutility. A recursive version has the Bellman equation:

J(x) = max min{u(v,x) +p(6wTw + EJ(x'))}

subject to the law of motion x' = Ax + Bv + C(w + e'). The value func-tion again takes the form of equation (25), so the Bellman equation canbe expressed as

-xTPx - q = max mm{-(vJQv + xTRx + 2vJSx) JV W

- /lE([Ax + Bv + C(w + s')]TP{Ax + Bv + CE') + p)}. (31)

The minimization leads to


zv=(0I- CTPC)~1CTP(Ax + Bv)

and

6wTw - (Ax + Bv + Cw)JP(Ax + Bv + Cw) = (Ax + Bv)TP(Ax + Bv),

where

P = P + 9~1PC(I-9~lCTPC)~lCJP. (32)

Comparing (32) with (29), we see that risk-sensitive and robust controllead to similar objective functions and produce identical decision rulesif 9~l = 2a.

A different representation of the problem leads to a solution that fitsexactly into the traditional optimal control framework and is thereforeamenable to traditional computational methods. The min-max theoremsuggests that we can compute the solutions for v and w simultane-ously. With this in mind, define:

Then the problem is one of optimal control and can be solved usingthe Riccati equation (27) applied to (Q,R, S, A, B). The optimal controlsare v = — F\X and w = — F2X, where the F, come from partitioning F. Adoubling algorithm applied to this problem provides an efficient com-putational technique for robust and risk-sensitive control problems.

Entropy constraints. As in the static case, dynamic robust controlproblems can be derived using an entropy constraint. Hansen and Sar-gent (2004, Chapter 6) suggest

t=Q

Discounting is convenient here, but is not a direct outcome of a multi-period entropy calculation. They argue that discounting allows distor-tions to continue to play a role in the solution; without it, the problemtends to drive It and wt to zero with time. A recursive version of theconstraint is

A recursive robust constraint problem is based on an expanded statevector, (x, I), and the law of motion for 7 above. As in the static case,


the result is a theory of the Lagrange multiplier 9. Conversely, the solu-tion to a traditional robust control problem with given 9 can be used tocompute the implied value of 1$. The recursive version highlights an in-teresting feature of this problem: nature not only minimizes at a pointin time, but also allocates entropy over time in the way that has thegreatest adverse impact on the agent.

Example 16 (robust precautionary saving) Consider a linear-quadraticversion of the precautionary saving problem. A theoretical agent hasquadratic utility, u(ct) = (ct — y)

2, and maximizes the expected dis-counted sum of utility subject to a budget constraint and an auto-regressive income processs:

at+i = r(at - ct) + yt+i

yt+i = (1 - (p)y + (pyt + tret+i,

where {st} ~ NID(0,1). We express this as a linear-quadratic controlproblem using ct as the control and (l,at, yt) as the state. The relevantmatrices are

Qs

L =

S T 1R.

-

(1(1

_

1

- 70

0

1 0

— (p)y r

-


making the actual and distorted dynamics

A =1

0.2

0.2

A-CF2 =

0 01.0526 0.8

0 Oi

1

-0.1247-0.1247

>

1

0

0

.0661

.01340

0

0

.8425

.8425

The distorted dynamics are pessimistic (the income intercept changesfrom 0.2 to —0.1247) and more persistent (the maximal eigenvalueincreases from 1.0526 to 1.1086). The latter calls for more aggressiveresponses to movements in a and y. [Adapted from Hansen, Sargent,and Tallarini (1999) and Hansen, Sargent, and Wang (2002).]

5.3 Beyond LQG

You might conjecture (as we did) that the equivalence of risk-sensitiveand robust control hinges critically on the linear-quadratic-gaussianstructure. It doesn't. The critical functional forms are the exponentialcertainty equivalent and the entropy constraint. With these two ingre-dients, the objective functions of risk-sensitive and robust control arethe same.

We demonstrate the equivalence of risk-sensitive and robust con-trol objective functions in a finite-state setting where the math is rel-atively simple. Consider an environment with conditional probabilitiesp(z'\z). Since z is immaterial in what follows, we ignore it from nowon. In a typical dynamic programming problem, the Bellman equationincludes the term EJ = J2z' P(Z')KZ')- A robust control problem has asimilar term based on transformed probabilities p(z') whose valuesare limited by an entropy penalty:

E

If p(z') = p(z'), this is simply EJ. The new elements are the minimiza-tion with respect to p (the defining feature of robust control), the en-tropy penalty on the choice of p (the standard functional form), and


the constraint that the transformed probabilities sum to 1. For eachp(z'), the first-order condition for the minimization is

J(z') + 0{log[p(z')/p(z')] + 1} + * = 0. (33)

If we multiply by p(z') and sum over z', we get J + # + 1 = 0, whichwe use to eliminate X below. Each first-order condition implies

p(z') = p(z')e-V

If we sum over z' and take logs, we get

]=-0\og(Yv(z')exp[-Kz')/e\Z •

our old friend the exponential certainty equivalent with risk aversionparameter a = 9~l. If we place / in its Bellman equation context, we'veshown that robust control is equivalent (even outside the LQG class) tomaximizing Kreps-Porteus utility with an exponential certainty equiv-alent. The log in the entropy constraint of robust control reappearsin the exponential certainty equivalent. An open question is whetherthere's a similar relationship between Kreps-Porteus preferences with(say) a power certainty equivalent and a powerlike alternative to theentropy constraint.

5.4 Discussion: Interpreting Parameters

Risk-sensitive and robust control raise a number of estimation issues,some we've seen and some we haven't. Risk-sensitive control is basedon a special case of Kreps-Porteus preferences and therefore leads tothe same identification issues we faced in the previous section: weneed variation over time in the conditional distribution of next period'sstate to distinguish time and risk parameters.

Robust control raises new issues. Risk-sensitive and robust controllead to the same decision rules, so we might regard them as equivalent.But they're based on different preferences and therefore lead to differ-ent interpretations of parameters. While risk-sensitive control suggestsa risk-averse agent, robust control suggests an agent who is uncertainabout the model that generated the data. In practice, the two can bequite different. One difference is plausibility: we may find an agentwith substantial model uncertainty (small 0) more plausible than onewith enormous risk aversion (large a). Similarly, if we find that amodel estimated for Argentina suggests greater risk aversion than one


estimated for the United States, we might prefer to attribute the differ-ence to model uncertainty. Hansen and Sargent (2004, Chapter 8) havedeveloped a methodology for calibrating model uncertainty (error de-tection probabilities) that gives the robust-control interpretation somedepth. Another difference crops up in comparisons across policy re-gimes: the two models can differ substantially if we consider policyexperiments that change the amount of model uncertainty.

6. Ambiguity

In Sections 3 and 4, agents know the probabilities they face, andwith enough regularity and repetition, an econometrician can estimatethem. Here we consider preferences when the consequences of ourchoices are uncertain or ambiguous. It's not difficult to think of suchsituations: what are the odds that China revalues this year by morethan 10%, that the equity premium is less than 3%, or that productivityshocks account for more than half of the variance of U.S. outputgrowth? We might infer probabilities from history or market prices,but it's a stretch to say that we know (or can find out) these probabil-ities, even though they may affect some of our decisions. One line ofattack on this issue was suggested by Savage (1954): that people maxi-mize expected utility using personal or subjective probabilities. In thiscase, we retain the analytical tractability of expected utility but lose theempirical convenience of preferences based on the same probabilitiesthat generate outcomes (rational expectations). Another line of attackgeneralizes Savage: preferences are characterized by multiple probabil-ity distributions, or priors. We refer to such preferences as capturingambiguity and ambiguity aversion, and explore two examples: Gilboaand Schmeidler's (1989) max-min expected utility for static environ-ments and Epstein and Schneider's (2003) recursive multiple priorsextension to dynamic environments. The central issues are dynamicconsistency (something we need to address in dynamic settings) andidentification (how do we distinguish agents with ambiguous prefer-ences from those with expected utility?).

6.1 Static Ambiguity

Ambiguity has a long history and an equally long list of terminology.Different varieties have been referred to as Knightian uncertainty,Choquet expected utility, and expected utility with nonadditlve (sub-


jective) probabilities. Each of these terms refers to essentially the samepreferences. Gilboa and Schmeidler (1989) provide a simple represen-tation and an axiomatic basis for a preference model in which an agententertains multiple probability models or priors. If the set of priors isII, preferences are represented by the utility function

U({c(z)}) = minVn{z)u[c(z)] = min Enu(c). (34)

Gilboa and Schmeidler refer to such preferences as max-min becauseagents maximize a utility function that has been minimized with re-spect to the probabilities n. We denote probabilities by n, rather thanp, as a reminder that they are preference parameters. The defining fea-ture is II, which characterizes both ambiguity and ambiguity aversion.If II has a single element, (34) reduces to Savage's subjective expectedutility.

Gilboa and Schmeidler's max-min preferences incorporate aver-sion to ambiguity: agents dislike consequences with unknown odds.Consider an agent choosing among mutually exclusive assets in athree-state world. State 1 is pure risk: it occurs with probability1/3. State 2 is ambiguous: it occurs with probability 1/3 — y, with—1/6 < y < 1/6. State 3 is also ambiguous and occurs with probability1/3 + y. The agent's probability distributions over y define n. Weuse the distributions ny{y = g) = 1 for -1/6 < g < 1/6, which imply(1/3,1/3—g, 1/3+g) as elements of n . These distributions over yare dogmatic in the sense that each places probability 1 on a partic-ular value. The approach also allows nondogmatic priors, such as7i7(y = —1/6) = 7z7(y = 1/6) = 1/2. In this setting, consider the agent'svaluation of three assets: A pays 1 in state 1, nothing otherwise; B pays1 in state 2; and C pays 1 in state 3. How much is each asset worthon its own to a max-min agent? To emphasize the difference betweenrisk and ambiguity, let u(c) = c. Using (34), we find that asset A isworth 1/3 and assets B and C are each worth 1/6. The agent is appar-ently averse to ambiguity in the sense that the ambiguous assets, Band C, are worth less than the unambiguous asset, A. In contrast, anexpected utility agent would never value both B and C less than A.

Example 17 (portfolio choice and nonparticipation) We illustrate the im-pact of ambiguity on behavior with an ambiguous version of Example6. An agent has max-min preferences with u(c) = ca/a and a = 0.5. Sheinvests fraction w of initial wealth a0 in a risky asset with returns


[r(l) —K\— a, r(2) — K\ + a, with a = 0.17] and fraction 1 — zv in arisk-free asset with return ro = 1.01 in both states. Previously weassumed the states were equally likely: n(l) = n{2) = 1/2. Here we letTT(1) take on any value in the interval [0.4,0.6] and set n(2) = 1 — n{l).Two versions of this example illustrate different features of max-minpreferences.

• Version 1: First-order risk aversion generates a nonparticipation re-sult. With expected utility, agents are approximately neutral to fairbets. In a portfolio context, this means they'll buy a positive amount ofan asset whose expected return is higher than the risk-free rate, andsell it short if the expected return is lower. They choose w — 0 onlyif the expected return is the same. With multiple priors, the agentchooses w = 0 for a range of values of K\ around the risk-free rate (thenonparticipation result). If we buy, state 1 is the worst state and themin sets n(l) — 0.6. To buy a positive amount of the risky asset, thefirst-order condition must be increasing at w = 0:

0.6(r0)^fa - a - r0) + 0 .4(r 0 )a -Vi + a - r0) > 0,

which implies K\-rQ> 0.2(7 or K\ > 1.01 + 0.2(0.17) = 1.044. If wesell, state

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Exotic Preferences for Macroeconomists · 2020. 3. 20. · 320 Backus, Routledge, & Zin (A) zi = 1 z2 = (zo, 1,1) 21=2 Figure 1 A representative event tree. This event tree illustrates